2.1. The area of a door opening is 9,5% more than the area of a bedroom window. Calculate, in metres, the width of the door opening.

The dimensions of the windows are $170 \text{ cm} \times 140 \text{ cm}$.
Convert centimeters to meters: $170 \text{ cm} = 1.70 \text{ m}$ and $140 \text{ cm} = 1.40 \text{ m}$.

$$\text{Area}_{\text{window}} = 1.70 \text{ m} \times 1.40 \text{ m} = 2.38 \text{ m}^2$$

The area of a door opening is $9.5\%$ more than the area of a bedroom window.

$$\text{Area}_{\text{door}} = \text{Area}_{\text{window}} \times (1 + 0.095)$$ $$\text{Area}_{\text{door}} = 2.38 \text{ m}^2 \times 1.095 = 2.6051 \text{ m}^2$$

The height of the door opening is $2.14 \text{ m}$. Using the formula Area = length $\times$ width (or height $\times$ width):

$$\text{Width}_{\text{door}} = \frac{\text{Area}_{\text{door}}}{\text{Height}_{\text{door}}}$$ $$\text{Width}_{\text{door}} = \frac{2.6051 \text{ m}^2}{2.14 \text{ m}} = 1.217336... \text{ m}$$

Rounding to two decimal places:

$$\text{Width}_{\text{door}} = \boxed{\text{}1.22 \text{ m}}$$

1.2.2 Verify whether Nikwe's statement is CORRECT.

The total surface area of the inside walls of the two rooms is $54.28 \text{ m}^2$.
She applies two coats of paint.

$$\text{Total area to paint} = 54.28 \text{ m}^2 \times 2 = 108.56 \text{ m}^2$$

The spread rate is $3 \text{ m}^2/\text{litre}$.

$$\text{Volume of paint needed} = \frac{\text{Total area to paint}}{\text{Spread rate}}$$ $$\text{Volume of paint needed} = \frac{108.56 \text{ m}^2}{3 \text{ m}^2/\text{litre}} = 36.186... \text{ litres}$$

Rounding to two decimal places:

$$\text{Volume of paint needed} \approx 36.19 \text{ litres}$$

Nikwe stated that she will need exactly 5 buckets. For this statement to be correct, 5 buckets must contain $36.19 \text{ litres}$ of paint. This would mean each bucket contains $36.19 \text{ litres} \div 5 = 7.238 \text{ litres}$. Since $7.238 \text{ litres}$ is not a standard, commonly available bucket size (e.g., 1L, 2.5L, 5L, 10L), Nikwe's statement is likely incorrect in a practical sense. If she buys 5 buckets of a common size (e.g., 5L), she will only have $25 \text{ litres}$, which is not enough.

$$\text{Nikwe's statement is } \boxed{\text{}INCORRECT}$$

1.2.3 Verify whether the invoice is CORRECT.

Normal labour charge: R75.90 per hour. He works 6 hours on Friday.

$$\text{Friday earnings} = 75.90 \text{ R/hour} \times 6 \text{ hours} = 455.40 \text{ R}$$

On Saturday, he charges $50\%$ more than the normal rate.

$$\text{Saturday hourly rate} = 75.90 \text{ R/hour} \times (1 + 0.50) = 75.90 \text{ R/hour} \times 1.50 = 113.85 \text{ R/hour}$$

He works 6 hours on Saturday.

$$\text{Saturday earnings} = 113.85 \text{ R/hour} \times 6 \text{ hours} = 683.10 \text{ R}$$

$$\text{Total charge} = \text{Friday earnings} + \text{Saturday earnings}$$ $$\text{Total charge} = 455.40 \text{ R} + 683.10 \text{ R} = 1138.50 \text{ R}$$

The painter's invoice is R1 238.
Since R1 138.50 $\neq$ R1 238, the invoice is not correct.

$$\text{The invoice is } \boxed{\text{}INCORRECT}$$